Testing CPT violation, entanglement and gravitational interactions in particle mixing with trapped ions

By analyzing the analogies between the effective system of $N$ spins described by the Ising Hamiltonian and the phenomenon of the self--gravity in mixed particle systems, we show that cooled ions held in a segmented ion trap and exposed to a magnetic field gradient can simulate the proposed mechanism of mutual interaction in mixed neutrino system. We show that with trapped ions one can reproduce the expected corrections to the flavor transitions and the $CPT$ violation induced by gravity on flavor fields, which may have played an important role in the early stages of the universe. The results presented are experimentally testable. They indicate that ions confined in microtraps can represent a new tool to test fundamental phenomena of nature.